The sine, cosine and tangent functions and their graphs, the sine and cosine rules, the area of a triangle, and exact values of trigonometric ratios for standard angles.

What this dot point is asking

OCR wants you to understand the sine, cosine and tangent functions and their graphs (including period, symmetry and key values), use the sine and cosine rules to solve any triangle, find the area of a triangle from two sides and the included angle, deal with the ambiguous case of the sine rule, and recall the exact values of the trigonometric ratios for the standard angles.

The answer

The three functions and their graphs

The functions $\sin\theta$ and $\cos\theta$ are periodic with period $360^\circ$ (or $2\pi$ radians), oscillating between $-1$ and $1$. The function $\tan\theta$ has period $180^\circ$ (or $\pi$) and vertical asymptotes where $\cos\theta = 0$, that is at $\theta = 90^\circ, 270^\circ, \dots$ Knowing the shape of each graph lets you read off how many solutions an equation has in a given interval and find all of them using the symmetry of the curve.

Exact values

You must know the exact ratios for $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$ without a calculator. They come from the half-equilateral triangle and the unit square.

The sine rule

The sine rule links each side of a triangle with the angle opposite it. Use it when you know two angles and any side, or two sides and a non-included angle.

The cosine rule

The cosine rule is the right tool when you know two sides and the included angle (to find the third side), or all three sides (to find an angle).

Area of a triangle

When you know two sides and the angle between them, the area is $\tfrac{1}{2}ab\sin C$, where $C$ is the included angle. This is faster and more accurate than finding a perpendicular height.

Examples in context

Choosing the right rule

The decision is mechanical. If you have an angle and the side opposite it, the sine rule works. If you have two sides and the angle between them, or three sides, use the cosine rule. Many multi-step questions need the cosine rule first to find a side, then the sine rule or the area formula to finish.

The ambiguous case

When the sine rule gives an angle from a side opposite a known acute angle, there can be two triangles: the calculator returns the acute value, but the obtuse supplement $180^\circ - \theta$ may also fit. Always check whether the obtuse option keeps the angle sum below $180^\circ$; if it does, both triangles are valid and the question usually asks for both.

Try this

Q1. A triangle has sides $4$ cm and $5$ cm with an included angle of $60^\circ$. Find its area. [2 marks]

• Cue. $\tfrac{1}{2}(4)(5)\sin 60^\circ = 10 \times \tfrac{\sqrt{3}}{2} = 5\sqrt{3} \approx 8.66$ cm$^2$.

Q2. In triangle $ABC$, $a = 8$, $b = 11$ and $c = 7$. Find angle $A$. [3 marks]

• Cue. $\cos A = \dfrac{11^2 + 7^2 - 8^2}{2(11)(7)} = \dfrac{106}{154} = 0.688$, so $A \approx 46.5^\circ$.